Although I am a mathematics education graduate student and am not required to take mathematics courses as a part of my PhD, I had the opportunity to take several mathematics graduate courses this summer as part of a Mathematics Graduate Certificate for Teachers program, which aims to bring mathematics graduate courses to current classroom teachers in order to certify them to teach college-level mathematics. I wanted to summarize some of the lessons I learned about teaching from these classes that I can bring into my own practice as a mathematics teacher. These fall into three categories: questioning, alignment, and metacognitive modeling.

**Questioning:** There’s an old saying that math instructors should “always answer a question with a question.” I had numerous opportunities to watch the ways in which instructors would respond to student questions. Direct answers tended to make the encounter about the teacher and taught the student that mathematics was about rules and procedures, while asking the student “what do you think?” kept the encounter student-centered and lead to more conceptual understandings. This semester, I am going to watch carefully what happens when I give different types of responses to students in order to fine tune the way in which I offer responses.

**Alignment:** Another thing that I had an opportunity to observe was the ways in which instructors structured their classes, particularly in the ways in which they aligned their lectures, in-class activities, homework, and examinations. One key take-away from this: giving problems that are completely different from what was done in-class and on the homework in order to assess student thinking on novel problems can backfire if you are primarily grading based on correctness or adherence to the way you as an instructor think the problem should be solved. It seems like there are two directions here; one is to give problems on tests that look a lot like what was done in the homework and the review, and two is to grade in ways that take into account students’ creative reasoning and attempts towards a solution. I generally go for the former, but am interested in experimenting more with the latter. In particular, I am thinking of using what Annette Leitze refers to as an analytic rubric, where each stage of the problem-solving process is graded separately.

**Metacognitive Modeling:** One key aspect of graduate classes in mathematics was watching instructors solving problems during lecture where they explicitly elucidate the decisions they were making, as in, *why did they do this and not that, what happens when you reach a dead end*, etc. A key observation here is that although it is important to model realistic problem solving in this way, doing too much of it can sometimes lead to a lack of student confidence in the instructor. It is okay for the instructor to be stuck once in a while to show students what that is like, but at the same time it is important to spend adequate time preparing for class and to occasionally work harder problems out ahead of time so that students are confident that you know the material that you are teaching.

It was invigorating to my practice to have the opportunity to take mathematics content courses at the graduate level, especially since as a mathematics education doctoral student, I ordinarily only have the opportunity to take courses on pedagogy, history, theory, and curriculum rather than actual mathematics content courses. I am excited to begin this next semester with new ideas and thoughts about how to approach my own teaching.

I am an online math tutor at Learn Zoe and I find it applicable to my field. I am glad I run across your website. I had a good time reading it. Thank you.